The original article was published in Fixed Point Theory and Applications 2013 2013:129

The assertion in [1] that Caristi’s theorem holds in generalized metric spaces isbased, among other things, on the false assertion that if{pn} is a sequence in a generalized metric space(X,d), and if {pn} satisfies ∑i=1∞d(pi,pi+1)<∞, then {pn} is a Cauchy sequence. In Example 1 below wegive a counter-example to this assertion, and in Example 2 we show that, infact, Caristi’s theorem fails in such spaces. We apologize for anyinconvenience.

For convenience we give the definition of a generalized metric space. The conceptis due to Branciari [2].

Definition 1 Let X be a nonempty set and d:X×X→[0,∞) a mapping such that for all x,y∈X and all distinct points u,v∈X, each distinct from x and y:

Therefore (X,d) is a generalized metric space. Now suppose{nk} is a Cauchy sequence in (X,d). Then if ni≠nk and d(ni,nk)<1, |ni−nk| must be odd. However, if {nk} is infinite, |ni−nk| cannot be odd for all sufficiently large i, k. (Suppose ni>nj>nk. If ni−nj and nj−nk are odd, then ni−nk is even.) Thus any Cauchy sequence in(X,d) must eventually be constant. It follows that(X,d) is complete and that {n} is not a Cauchy sequence in (X,d). However, ∑i=1∞d(i,i+1)<∞.

Theorem 2 of [1] asserts that the analog of Caristi’s theorem holds in acomplete generalized metric space (X,d). Thus a mapping f:X→X in such a space should always have a fixed pointif there exists a lower semicontinuous function φ:X→R+ such that

d(x,f(x))≤φ(x)−φ(f(x))for each x∈X.

The following example shows this is not true in the space described inExample 1.

Example 2 Let (X,d) be the space of Example 1, letf(n)=n+1 for n∈N, and define φ:N→R+ by setting φ(n)=2n. Obviously f has no fixed points and,because the space is discrete, φ is continuous. On the other hand, f satisfies Caristi’s condition:

12n=d(n,f(n))≤φ(n)−φ(f(n))=2n−2n+1.

To see this, observe that

12n≤2n−2n+1=2n(n+1).

This is equivalent to the assertion that

2n+1≥n(n+1).

(C)

The proof is by induction. Clearly (C) holds if n=1 or n=2. Assume (C) holds for some n∈N, n≥2. Then

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